The different concentrations of SWCNTs in polymer matrix are studied for their current versus voltage properties using electrode slides with an electrode gap of 50µm (chapter 3, section 3.2). All the material composites showed a non-linear behaviour in terms of current versus voltage measurements. At lower concentrations, such as 0.1%, the current was measured in nA, but as the concentration of SWCNTs was increased in the polymer matrix, the current measurements were recoded in mA. This behaviour can be attributed to the fact that, the increased concentration of SWCNTs led to increased interconnections between SWCNTs bundles. This behaviour is also noted in viscosity measurements of these composites, as described in the previous section. The comparison of current versus SWCNTs
concentration is shown in Figure 6.3. The log of current versus SWCNTs concentration is linear up to 1.0% SWCNTs concentration, but this gradient reduces after 1.0% concentration. This behaviour can be attributed to the electrical percolation threshold of the SWCNTs network, where the voltage will become less affected by the increased concentration of SWCNTs. The electrical percolation threshold for SWCNTs network, as reported in literature [175], is between 0.17 − 0.70% and several other factors such as a polymer, the source of SWCNTs and additional purification also affect the percolation threshold of SWCNTs networks. This percolation threshold for current experiments is in accordance to what is reported in literature [175], however, the polymer used for current experiments is PBMA (butyl group instead of methyl group), which has longer chains length and aids in better suspension and formation of thin films. In order to understand the
Figure 6.3: Comparison of current and voltage for various SWCNTs concentrations [3]
conduction mechanism of SWCNTs/polymer composites, the logarithm of current versus voltage were plotted, which indicated that at higher concentrations the
current was directly proportional to voltage with a slope 1. However, at lower concentration of SWCNTs the slope was recoded as 1.5, which means that there are other mechanisms at work with the SWCNTs network. It was suggested that this could be the Poole-Frankel (PF) effect [176], which is widely reported in literature as a conduction mechanism in Carbon nanotubes network. Hence, the current versus voltage relationship can be written as:
I ∝ V exp − Ed− βP FV 1 2 kT ! (6.2.3) where βP F = e3 π0rd !12 (6.2.4) The Poole-Frankel fit for two concentrations of SWCNTs (i.e. very low: 0.11% and very high: 3.20%) is plotted in Figure 6.4. A linear response with gradient 0.46 is observed in lower concentration of SWCNTs. However, higher concentration of SWCNTs has not showed a linear response between current and voltage relationship which suggested that Poole-Frankel is not the dominant conduction mechanism in these networks of films.
6.3
Results and discussion
The SWCNTs/PBMA based material posses complex electrical characteristics, with a field dependent conductivity when the concentration of SWCNTs is below a certain threshold. Using this data various concentrations listed in Table 3.2 in chapter 3, are studied for computer controlled optimisation of logic gates/circuits using the hardware described in Chapter 3, Section 3.3. The problem formulation is same as described in chapter 5, section 5.2 and AND, OR and half adder circuits are solved.
Figure 6.4: Poole-Frankel fit for low (0.11%) and high (3.20%) concentrations of SWCNTs [3]
The two parameters, i.e. fitness function value (0 as best value) and the number of function evaluations to achieve the fitness value are used to judge the suitability of the material for the computational problem. The number of function evaluations to reach the desired fitness value provides an indication that how well the material’s properties are used for performing the target computation. The less number of function evaluations to reach an optimal solution indicates a more flexible material and an efficient hardware-in-loop optimisation. These investigations have been performed by keeping all the parameter values of Nelder-Mead algorithm the same for all the material composite. Table 6.2 provides the values of the number of average number of function evaluations achieved during 5 different optimisation runs by each material composite to reach the solution. It can be seen that the material with SWCNTs concentration above 0.1% require less number of function evaluations to converge to optimal solution, whereas, the concentration less that 0.1% required a larger number of function evaluations and reached the
Concentration of SWCNTs (wt%)/PBMA
Average no. of Function Evaluations (F Eavg) using NM
OR AND Half Adder F Eavg F Eavg F Eavg
0.11 1881 1889 1788 0.25 1115 1881 1998 0.51 1228 1258 1547 0.74 1228 1717 1587 0.99 22 45 142 1.49 39 32 285 2.39 21 45 118
Table 6.2: Average number of function evaluations during 5 different runs to train various concentrations of SWCNTs/PBMA composites for AND, OR and Half Adder circuit
termination criteria (maximum number of iterations allowed) without achieving the desired fitness function value. The number of function evaluations performed by the Nelder-Mead algorithm depends upon the simplex’s expansion, contraction or shrunk rate during each iteration.
The fitness function value indicates the success of reaching the optimal solution for approximating the logic gate/circuits, with 0 being the desired value. The minimum (Hmin), maximum (Hmax) and average (Havg) of fitness values achieved during optimisation process and the test accuracy (Φavg) achieved by various
concentrations of SWCNTs/PBMA composites for three different logic gates/circuits are provided in Table 6.3. These values are achieved during 5 different runs for every material sample. In a similar manner to number of function evaluations,
SWCNTs (wt%)/ PBMA
Nelder Mead Optimisation results from 5 different runs
OR AND Half Adder
H min H max H avg Φavg H min H max H avg Φavg H min H max H avg Φavg 0.11 12 23 22 25% 4 6 6 22% 14 15 14 15% 0.25 9 11 9 28% 3 4 4 35% 17 14 17 15% 0.51 9 12 9 23% 6 7 8 48% 9 11 9 12% 0.74 4 5 4 48% 1 3 3 45% 1 3 3 55% 0.99 0 0 0 100% 0 0 0 100% 0 0 0 100% 1.49 0 0 0 100% 0 0 0 100% 0 0 0 100% 2.39 0 0 0 100% 0 0 0 100% 0 0 0 100%
Table 6.3: Fitness function values for various concentrations of SWCNTs/PBMA composites for AND, OR and Half adder circuit
the material composites having concentration of 1.0% and above achieved a fitness value of 0 for all three types of logic gates/circuits. Where as, at lower concentrations the fitness values are greater than zero indicating that the algorithm failed to converge to an optimal solution for all three types of logic gates/circuits.
It can be seen in Table 6.2 and Table 6.3 that AND and OR gate require less number of function evaluations for the material composites having SWCNT concentration above 1.0%. Whereas, the solution to half adder requires more iterations as compare to AND and OR gate, for the material composites having a concentration above 1.0%. This is attributed to the fact that in the half-adder circuit the optimisation algorithm is training the material to behave as two logic gates at the same time (an AND and XOR), for the same inputs. This increases the complexity and hence require more function evaluation to reach an optimal solution. It should be noted that threshold of 1.0% concentration of SWCNTs is
retained for the solution of OR, AND and a Half adder circuit. Hence, a denser and randomly dispersed network of SWCNTs in the polymer matrix (PBMA) is more favourable to training logic gates/ circuits, following the threshold scheme. The concentration of SWCNTs is directly proportional to the point where the conductivity starts increasing. Below this threshold concentrations, there are not enough connections within the SWCNTs/polymer matrix from which a meaningful computation can be extracted.
In some experiments performed by Mohid et al. [112] it was reported that the material sample with (1.0% SWCNT/PBMA) was successfully trained to solve even parity [116], tone discrimination [114], Robot controller [117] and data classification problems [115]. These investigations have also reported that the material samples with concentration lower than 0.02% in polymer (PMMA) cannot be evolved to solve the computational problem, however the ratio of PMMA varied. It was also observed that no evolution is possible when there was no material was present on board. These experiments also supported that the 1.0% SWCNT concentration in fixed polymer ratio (PBMA) are suitable to studied further for EIM and also other materials with similar physical and conductive properties can also be investigated for this study.
6.4
Conclusions
The work discussed in this chapter described the relationship between the concentration of SWCNTs in PBMA matrix and its effect on training the composite for solution of a computational problem. The SWCNTs/PBMA composites possess complex electrical and mechanical properties that can be used for unconventional computing. The study of electrical and mechanical properties of these composites showed a percolation threshold of 1.0%, where the electrical and mechanical properties of
the composites changed, which resulted in sheer thinning behaviour and change in electrical conductivity of the composites. Above the percolation threshold the rate of increase of conductivity is reduced and at lower concentrations, there are less number of interconnections between SWCNTs network and Poole-Frankel provided a good fit to data.
The various concentrations of SWCNTs/PBMA composites have been used to solve logic gate/circuit problem, using a specially designed hardware platform. A certain threshold in terms of SWCNTs concentration is observed for the solution of Logic gates/circuits. The threshold logic concept to interpret the outputs from the material as logic gate was used in conjunction with Nelder-Mead algorithm, which provided the values for optimal voltages. The results of these experiments showed that a threshold of 1.0 wt% SWCNTs is most suitable for solving logic gate/circuit problem. This is similar to what has been observed during the study of electrical and mechanical properties of these material composites.
These experiments are different from data classification and frequency classification problems [115], [177], where varying concentrations of SWCNTs as well as varying concentrations of PMMA were used to solve these problems. No conclusion was drawn from these experiments as to show if the concentration of SWCNTs matters in the polymer ratio to solve these computational problems.
These results provided a clear indication of the link between the SWCNTs concentration and the ability to solve a simple computational problem. The further investigation of these material composites to solve other computational problems will provide more insight into this relationship.
Training SWCNTs/PMMA
composites to solve complex logic
circuits using Particle Swarm
algorithm on Mecobo
This chapter presents the results of using a particle swarm optimisation (PSO) algorithm for evolving complex logic circuits in SWCNTs based composites on a purpose-built platform, Mecobo (version 4.1) (Chapter 3, Section 3.4). The material used is a composite of SWCNTs, dispersed randomly in a polymer(PMMA) forming a complex conductive network. Following the EIM methodology, the conductance of the material is manipulated for evolving complex threshold-based logic circuits.
The results of experiments in chapter 5, section 5.3.5 showed that the choice of contact points for the application of incident signals (input and configuration signals) and the collection of output signals have a significant effect on the computation performed by the SWCNTs/PMMA composites. The Mecobo platform is flexible
enough to put the choice of input and output signals under optimisation control. Hence, the experiments described in this chapter use a different approach from previous experiments for solving logic gates problem describe in chapter 5 and chapter 6. The material training problem is formulated as a constrained, mixed integer optimisation problem. The problem is solved using PSO in conjunction with the shortest position value rule. The results showed that the conductive properties of SWCNTs can be used to configure these materials to evolve multiple input/ output logic circuits using a more flexible hardware.
7.1
Introduction
The EIM methodology requires a platform that provides access to physical properties of the material in use and can bridge the gap between the analogue nature of the material and the digital nature of a computer supervising an evolutionary search. The experiments described in chapter 5, section 5.3.5, [178], and chapter 6, an mbed platform was used for evolving threshold logic circuits using population based optimisation.
The hardware used for those studies was relatively inflexible and did not allow the use of algorithms with extended vectors of decision variables regarding the selection of an incident signal’s location of application on the material body. In comparison, Mecobo is a more powerful and versatile platform which allows for a more flexible problem formulation to be realised. The Mecobo board can interface with a large variety of materials and also has the flexibility to control and map the variety of input, output and configuration signals and their properties. In addition to the different hardware used, this chapter extends the work reported in [178], [3] and [179] by using a Particle Swarm Optimisation (PSO) instead of the Nelder-Mead and Differential Evolution algorithms.
The material in use has SWCNTs that are randomly distributed forming an inhomogeneous random network of nanotube bundles. This material is spread over the micro-electrode arrays and the input/output locations can be arbitrary. The choice of input/output and configuration electrode terminals are left to be decided by the optimisation algorithm.
In order to achieve this, a pin routing module is placed between signal generating modules and the sampling buffer. Hence, in contrast to previous experiments, where pin configuration was predetermined, the experiments presented in this chapter implement variable pin configuration that is under the control of the optimisation algorithm.
Different computational problems have been suggested for such a system but for initial experiments, the calculation of Boolean functions based on threshold logic is considered here.